64 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
On comparing; this equation with (474.) we shall see that they are precisely identical. 
Whence we infer that an arc of the elliptic lemniscate is equal to an arc of a sphe- 
rical ellipse which is self-supplemental. It is very remarkable, that, whatever be the 
ratio of a to b the semiaxes of the plane ellipse or of the elliptic lemniscate, the arc 
is always equal to an arc of this particular species of spherical ellipse. 
There is another property of this spherical ellipse, that its area, together with 
twice the lateral surface of the cone, is equal to a hemisphere. See Theory of 
Elliptic Integrals, &c., p. 21. 
XCI. We may obtain under another form an expression for the arc of an elliptic 
lemniscate. 
Let the polar angle a be measured from the minor axis of the curve. Its equation 
in this case will be 
r 2 =a 2 sin 2 X-f-& 2 cos 2 X, 
ds 2 a 4 sin 2 A + b 4 cos 2 A 
dx 2 a 2 sin 2 A + Id cos 2 A 
b 2 
Assume tanX =^2 tan-4/, (475.) 
hence 
integrating, 
ds 2 a 9 b 2 dx a% 2 
dX 2 a 2 cos 2 \|/ + U l sin 2 \{/’ dfy a 4 cos 2 4> -f b 4 sirr-f/’ 
c # 
fa 4 — b 4 \ 
K « 4 ) 
sin 2 \f/ 
a/ 1 - 
fa 2 — b 2 \ 
l « 2 J 
sin 9 4/ 
. • (476.) 
Let, as before, a cylinder be erected on the ellipse and the sphere described from its 
centre with a radius equal to Vcr-^b 2 , it will cut the cylinder in a spherical ellipse, 
whose arc is given by the integral 
s tan 1 3 
k tana 
i l tan 2 a — tan 2 /3 \ . 0 , n / /sin 2 a — sin/3\ . 
J L 1 - ( fcuA ) sm ^JV 1 - ( 8 hA ) 81n ' + 
Now since 
•ft 2 + Z» 2 ’ 
sin 2 /3= 
b 2 
a 2 + b 2 
, tan/3 . „ b 3 tan 2 a — tan 2 /3 a 4 —b 4 sin 2 a — sin 2 /3 a 2 —b 2 
tana a 25 tan 2 a a 4 5 sin 2 a a 2 : 
substituting, we obtain s= 
c # 
J[>- 
(a 4 -b 4 \ 
\ a * J 
sin 2 \f/ 
\J i- 
(a 2 —b 2 \ . 2 
a , jsm«+ 
Now this is precisely the same equation as (476.), whence we infer that the arc of 
the elliptic lemniscate is equal to an arc of a self-supplemental spherical ellipse. 
Writing m for the parameter in this expression, we can easily show that the para- 
meters in this and the preceding formula (474.) are conjugate parameters. The con- 
